Weighted iterated Hardy-type inequalities

TL;DR

This paper establishes new equivalence and characterization theorems for the boundedness of weighted Hardy-type inequalities involving quasilinear operators in Lebesgue spaces, focusing on monotone function cones.

Contribution

It introduces novel equivalence theorems and characterizations for the boundedness of Hardy-type inequalities with quasilinear operators on monotone function cones.

Findings

New equivalence theorems for operator boundedness

Characterizations of weighted Hardy-type inequalities

Results on cones of monotone functions

Abstract

In this paper a reduction and equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. New equivalence theorems are obtained for the operator $T$ to be bounded in weighted Lebesgue spaces restricted to the cones of monotone functions, which allow to change the cone of non-decreasing functions to the cone of non-increasing functions and vice versa not changing the operator $T$. New characterizations of the weighted Hardy-type inequalities on the cones of monotone functions are given. The validity of so-called weighted iterated Hardy-type inequalities are characterized.